Linear Inviscid Damping and Vorticity Depletion for Shear Flows

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the same decay estimate of the velocity as the monotone shear flows. Moreover, we confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which along with the vorticity mixing leads to the damping for the base flows with stationary streamlines.